Asymptotic convergence of evolving hypersurfaces
If \psi\colon M^n\to \mathbb{R}^{n+1} is a smooth immersed closed hypersurface, we consider the functional \mathcal{F}_m(\psi) = \int_M 1 + |\nabla^m \nu |^2 \, d\mu, where \nu is a local unit normal vector along \psi , \nabla is the Levi-Civita connection of the Riemannian manifold (M,g) , with g t...
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| Published in | Revista matemática iberoamericana Vol. 38; no. 6; pp. 1927 - 1944 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
European Mathematical Society Publishing House
01.11.2022
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| Subjects | |
| Online Access | Get full text |
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| Summary: | If \psi\colon M^n\to \mathbb{R}^{n+1} is a smooth immersed closed hypersurface, we consider the functional \mathcal{F}_m(\psi) = \int_M 1 + |\nabla^m \nu |^2 \, d\mu, where \nu is a local unit normal vector along \psi , \nabla is the Levi-Civita connection of the Riemannian manifold (M,g) , with g the pull-back metric induced by the immersion and \mu the associated volume measure. We prove that if m>\lfloor n/2 \rfloor then the unique globally defined smooth solution to the L^2 -gradient flow of \mathcal{F}_m , for every initial hypersurface, smoothly converges asymptotically to a critical point of \mathcal{F}_m , up to diffeomorphisms. The proof is based on the application of a Łojasiewicz–Simon gradient inequality for the functional \mathcal{F}_m . |
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| ISSN: | 0213-2230 2235-0616 |
| DOI: | 10.4171/RMI/1317 |